Abstract

This project develops the theory for identification of dynamic stochastic general equilibrium (DSGE) models and studies its implications for estimation and inference. DSGE models have now reached the level of sophistication to permit analysis of important macroeconomic issues. Whereas the parameters in these models used to be calibrated, numerical advances in the last two decades have made it possible to estimate models with as many as a hundred parameters. Researchers are, however, aware that not all the parameters of interest can be consistently estimated because of identification problems: that different structural parameters can lead to indistinguishable outcomes. In spite of the recognition of this identification issue, a procedure has yet to be developed that tells us in a systematic manner how many parameters are identifiable, and if so which ones.

The first goal of this project is to study conditions under which a DSGE model is identifiable. The analysis is nonstandard as classical conditions for identification rely on assumptions that do not generally hold in DSGE models. Our proposal is to use the tight structure of DSGE models in order to establish new rank and order conditions for identification. Such conditions have not yet been proposed in the literature. T he focus is on covariance stationary process and the first step is to make precise the sense in which two dynamical systems can have observationally equivalent spectral densities. Adapting results from control theory, it is shown that the Markov (impulse response) parameters and the error variance of equivalent DSGE models must be related through a similarity transformation. These restrictions are then used to establish conditions under which DSGE models are identifiable from the spectrum of the observables. We will show that a DSGE model is identifiable even when its reduced form parameters are not. Formal identification conditions will be developed to explicitly handle the presence of measurement errors.

The second goal of this project is to study the estimation of DSGE model parameters. It is a well known fact that parameters that are not identifiable cannot be consistently estimated. This has important implications for both frequentist as well as Bayesian analysis as local non-identification leads to strange behavior of posteriors when flat priors are used. In spite of the importance of this problem, the literature on full information estimation of non-identified DSGE models is relatively small. The result that DSGE reduced form parameters are not identifiable from the spectrum has important implications for limited information estimation which uses only a subset of the autocovariances. The project will provide a complete characterization of an 'identifiable reduced form' consisting of dynamic equations and identities. The challenge for both full and limited information estimation is an error variance of reduced rank. The project will develop new estimation methods for singular systems without throwing away information or adding stochastic errors.

The third goal of the project is to focus on inference in DSGE models. Partially identified models pose challenging problems for testing. The project focuses on two issues. First, how to test statistical hypothesis in singular systems such as DSGE models when the parameters of interest are point identified but the nuisance parameters are not. Second, how to conduct inference in dynamic and possibly singular models when the parameters of interest are themselves only set identified. These problems are challenging, but are also relevant outside of the DSGE framework.

Intellectual Merit: Currently, there exists no formal identification results for DSGE models that allow for fewer shocks than endogenous variables. This project provides easy to evaluate rank and order conditions for identification that practitioners can check. Estimation and inference of structural parameters when the reduced form parameters are not identified, and set identification in singular systems are both new research topics. The results will be a new contribution to econometric theory.

Broader Impacts: DSGE models are increasingly used in policy analysis, so the impact of this work goes beyond a better methodological understanding of linear dynamical systems. The work is also relevant to estimation and inference of other singular models such as demand systems. Inference of partially identified dynamic models is of general interest. The computer code will be made available to the scientific community for non-commercial, research, and educational purposes.